5. INNER PRODUCT - Properties
Theorem 1: Let u, v, and w be vectors in Rn, and let c
be a scalar. Then
a. u•v = v•u
b. (u + v)•w = u•w + v•w
c. (cu)•v = c(u•v) = u•(cv)
d. u•u ≥0, and =0 iff u = 0
Properties (b) and (c) can be combined several
times to produce the following useful rule:
(c1u1 + c2u2 + … + cnun)•v = c1(u1•v) + … + cp(up•v)
12. ORTHOGONAL COMPLEMENTS
Theorem 3: Let A be an mxn matrix. The orthogonal
complement of the row space of A is the null space of
A, and the orthogonal complement of the column
space of A is the null space of AT:
Proof: ai = ith row of A
Row(A) = Span{rows of A}
Nul(A) = x:Ax = 0
ai • x = 0 for every i
x is orthogonal to each row of A
(Row ) NulA A
(Col ) Nul T
A A